I need help to find the finite distribution (fidi) of a homogeneous Poisson point process for overlapping set. For example:
Let $\Phi$ be a uniform Poisson point process of intensity $\lambda$ on $\mathbb{R}$. I want to determine $\mathbb{P}(\Phi(B_1) = n_1, \Phi(B_2) = n_2)$ for $B_1 = [0,2]$ and $B_2 = [1,3]$. Here, $n_1$ and $n_2$ are non-negative integers.